We report on the nonequilibrium spin noise of a single InGaAs quantum dot charged by a single hole under strong driving by a linearly polarized probe light field. The spectral dependency of the spin noise power evidences a homogeneous broadening and negligible charge fluctuations in the environment of the unbiased quantum dot. Full analysis of the spin noise spectra beyond the fluctuation-dissipation theorem yields the heavy-hole spin dynamics as well as the trion spin dynamics. Moreover, the experiment reveals an additional much weaker noise contribution in the Kerr rotation noise spectra. This additional noise contribution has a maximum at the quantum dot resonance and shows a significantly longer correlation time. Magnetic-field-dependent measurements in combination with theoretical modeling prove that this additional noise contribution unveils a charge reoccupation noise which is intrinsic in naturally charged quantum dots.